By Giuliano Sorani
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It is often hard, or impossible, to find closed form expressions for the inferred PDF of xt . Especially the normalizing factor p(yt |Yt−1 ) is usually troublesome. Linear models with Gaussian noise is one class of models where an analytical solution is known, the Kalman filter . The Kalman filter is further elaborated on in Chapter 4. In general conjugate priors, render manageable analytical results. The family of exponential distributions, of which the Gaussian distribution is a member, contains its own conjugate priors .
However, if the nominal trajectory is not known, or not accurate enough to allow for the higher order terms in the linearization to be discarded, other methods are needed. 2 Extended Kalman Filter In the extended Kalman filter (EKF) the problem with the lack of a nominal trajectory is solved using the information available from estimates by the filter itself. 8b) =:ut with T Ft := ∇x f (x, 0) x=ˆ xt|t T , Gt := ∇w f (ˆ xt|t , w) and w=0 , T Ht := ∇x h(x) x=ˆ xt|t−1 . For this linearization, K and P cannot be computed at design time, because neither x ˆt|t nor x ˆt|t−1 are available beforehand, both become available before they are needed in the recursion.
Studying the contour plot shows that in an information perspective most information is available if about 30–40% of the samples are outliers, and the outliers have substantially larger variance. The Kullback divergence between the outliers distributions, p1 (ω, k), and the normalized Gaussian distribution N (0, 1), has been computed to illustrate the difference between the distributions. 3(b). The relative accuracy and the Kullback divergence behaves similarly. 21) µ where 0 < Σ < ω1 − 1−ω and 0 < ω < 1 to get a proper distribution.